---
tag: EKXQFD7EMF
kind: concept
name: Busemann G-space
msc: [53C70, 51F99]
specializes: [0VKD1TP7F3]
sources:
  - url: https://en.wikipedia.org/wiki/Busemann_G-space
same_as:
  wikidata: Q55606213
  wikipedia: Busemann G-space
---

## Summary

A Busemann G-space is a [[metric space|0VKD1TP7F3]] in which shortest paths exist and extend a little past their ends in exactly one way.
These axioms, due to Herbert Busemann, capture the behaviour of [[geodesics|NSYMQMHPDN]] with no smoothness assumed.
Complete [[Riemannian manifolds|7CBT4SM83Y]] and many [[Finsler manifolds|5R6QZ388BB]] are examples, so G-spaces give a common synthetic frame for their geometry.

## Explanations

### L1 human

In a G-space you can always walk from one point to another along a shortest route.
You can also keep walking straight a bit past the end, and there is only one way to do so.
The round sphere has this property: a great circle arc continues as a unique great circle.
The plane with the taxicab distance fails it, because a shortest route can be continued straight on or bent at a corner with the same length.